Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 144*x^12 - 1440*x^10 + 4392*x^8 + 31104*x^6 - 38880*x^4 - 171072*x^2 + 187272)
gp: K = bnfinit(y^16 - 144*y^12 - 1440*y^10 + 4392*y^8 + 31104*y^6 - 38880*y^4 - 171072*y^2 + 187272, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 144*x^12 - 1440*x^10 + 4392*x^8 + 31104*x^6 - 38880*x^4 - 171072*x^2 + 187272);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 144*x^12 - 1440*x^10 + 4392*x^8 + 31104*x^6 - 38880*x^4 - 171072*x^2 + 187272)
\( x^{16} - 144x^{12} - 1440x^{10} + 4392x^{8} + 31104x^{6} - 38880x^{4} - 171072x^{2} + 187272 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(321236373250909071617512439808\)
\(\medspace = 2^{79}\cdot 3^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(69.85\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2^{2849/512}3^{3/4}\approx 107.8718783637264$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{6}a^{6}$, $\frac{1}{6}a^{7}$, $\frac{1}{18}a^{8}$, $\frac{1}{18}a^{9}$, $\frac{1}{18}a^{10}$, $\frac{1}{36}a^{11}$, $\frac{1}{756}a^{12}+\frac{1}{42}a^{10}-\frac{1}{42}a^{8}-\frac{1}{21}a^{6}-\frac{3}{7}a^{2}-\frac{1}{7}$, $\frac{1}{756}a^{13}-\frac{1}{252}a^{11}-\frac{1}{42}a^{9}-\frac{1}{21}a^{7}-\frac{3}{7}a^{3}-\frac{1}{7}a$, $\frac{1}{11131330392756}a^{14}-\frac{6414214859}{11131330392756}a^{12}+\frac{130975895}{11741909697}a^{10}+\frac{15970743212}{927610866063}a^{8}-\frac{4213755785}{88343892006}a^{6}-\frac{12903357547}{103067874007}a^{4}-\frac{36854398904}{103067874007}a^{2}-\frac{3845713902}{14723982001}$, $\frac{1}{189232616676852}a^{15}-\frac{2109539933}{11131330392756}a^{13}-\frac{10845247079}{798449859396}a^{11}+\frac{8591812745}{2252769246153}a^{9}-\frac{589007606533}{10512923148714}a^{7}+\frac{193232390467}{1752153858119}a^{5}-\frac{51578380905}{1752153858119}a^{3}+\frac{2527966688}{1752153858119}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{89991397}{5565665196378}a^{14}-\frac{90513313}{3710443464252}a^{12}+\frac{8974738}{3913969899}a^{10}+\frac{8339354203}{309203622021}a^{8}-\frac{1396358378}{44171946003}a^{6}-\frac{177245758673}{309203622021}a^{4}-\frac{33753362484}{103067874007}a^{2}+\frac{27528815009}{14723982001}$, $\frac{11560}{549858249}a^{14}+\frac{1361}{61095361}a^{12}-\frac{20974}{6960231}a^{10}-\frac{36656225}{1099716498}a^{8}+\frac{3443808}{61095361}a^{6}+\frac{43431304}{61095361}a^{4}-\frac{2946848}{61095361}a^{2}-\frac{74069793}{61095361}$, $\frac{1075871}{1855221732126}a^{14}+\frac{189734959}{11131330392756}a^{12}-\frac{72000}{1304656633}a^{10}-\frac{352967546}{103067874007}a^{8}-\frac{2691904872}{103067874007}a^{6}+\frac{7118318414}{103067874007}a^{4}+\frac{89704677252}{103067874007}a^{2}+\frac{80659713193}{103067874007}$, $\frac{14022713671}{189232616676852}a^{15}+\frac{259569179}{2782832598189}a^{14}+\frac{143135459}{927610866063}a^{13}+\frac{1621140953}{5565665196378}a^{12}-\frac{96759299}{9505355469}a^{11}-\frac{101342011}{7827939798}a^{10}-\frac{1345122609659}{10512923148714}a^{9}-\frac{323097601165}{1855221732126}a^{8}+\frac{123714761013}{3504307716238}a^{7}-\frac{48548631199}{618407244042}a^{6}+\frac{11235647917373}{5256461574357}a^{5}+\frac{983955881648}{309203622021}a^{4}+\frac{3954642275006}{1752153858119}a^{3}+\frac{436282820563}{103067874007}a^{2}-\frac{7313963052481}{1752153858119}a-\frac{841384175617}{103067874007}$, $\frac{14022713671}{189232616676852}a^{15}+\frac{259569179}{2782832598189}a^{14}-\frac{143135459}{927610866063}a^{13}+\frac{1621140953}{5565665196378}a^{12}+\frac{96759299}{9505355469}a^{11}-\frac{101342011}{7827939798}a^{10}+\frac{1345122609659}{10512923148714}a^{9}-\frac{323097601165}{1855221732126}a^{8}-\frac{123714761013}{3504307716238}a^{7}-\frac{48548631199}{618407244042}a^{6}-\frac{11235647917373}{5256461574357}a^{5}+\frac{983955881648}{309203622021}a^{4}-\frac{3954642275006}{1752153858119}a^{3}+\frac{436282820563}{103067874007}a^{2}+\frac{7313963052481}{1752153858119}a-\frac{841384175617}{103067874007}$, $\frac{159742409}{27033230953836}a^{15}-\frac{95492927}{5565665196378}a^{14}-\frac{8190961}{795095028054}a^{13}-\frac{1083040667}{11131330392756}a^{12}-\frac{15882419}{19010710938}a^{11}+\frac{13595510}{3913969899}a^{10}-\frac{24867303955}{4505538492306}a^{9}+\frac{61520519279}{1855221732126}a^{8}+\frac{1346127444}{250307694017}a^{7}-\frac{8596816138}{103067874007}a^{6}+\frac{153101794489}{750923082051}a^{5}-\frac{84862922268}{103067874007}a^{4}+\frac{144180138474}{250307694017}a^{3}-\frac{71900687109}{103067874007}a^{2}+\frac{83393693156}{250307694017}a+\frac{162985404901}{103067874007}$, $\frac{6115672733}{94616308338426}a^{15}+\frac{61174889}{11131330392756}a^{14}+\frac{4605031}{44171946003}a^{13}-\frac{3432418007}{11131330392756}a^{12}+\frac{660194113}{66537488283}a^{11}-\frac{4548909}{2609313266}a^{10}+\frac{2513818603223}{31538769446142}a^{9}+\frac{10355840188}{309203622021}a^{8}-\frac{2721799823444}{5256461574357}a^{7}+\frac{8802489727}{14723982001}a^{6}-\frac{13979478373874}{5256461574357}a^{5}+\frac{190289247955}{309203622021}a^{4}+\frac{9530793946294}{1752153858119}a^{3}-\frac{914093469841}{103067874007}a^{2}+\frac{43014451951182}{1752153858119}a-\frac{291265906807}{14723982001}$, $\frac{32604372077}{94616308338426}a^{15}-\frac{2797529333}{11131330392756}a^{14}+\frac{2053178560}{2782832598189}a^{13}-\frac{2432185969}{3710443464252}a^{12}-\frac{9508422298}{199612464849}a^{11}+\frac{810668893}{23483819394}a^{10}-\frac{897481706585}{1501846164102}a^{9}+\frac{841642906177}{1855221732126}a^{8}+\frac{951458402932}{5256461574357}a^{7}+\frac{1135958934}{14723982001}a^{6}+\frac{54848626769627}{5256461574357}a^{5}-\frac{2450250551881}{309203622021}a^{4}+\frac{15891798560902}{1752153858119}a^{3}-\frac{1376100886653}{103067874007}a^{2}-\frac{56115613566682}{1752153858119}a+\frac{173241442853}{14723982001}$, $\frac{19398505999}{189232616676852}a^{15}+\frac{386508805}{5565665196378}a^{14}+\frac{96966863}{1590190056108}a^{13}+\frac{174884113}{1236814488084}a^{12}-\frac{5909617321}{399224929698}a^{11}-\frac{112622753}{11741909697}a^{10}-\frac{4947900166921}{31538769446142}a^{9}-\frac{73891340657}{618407244042}a^{8}+\frac{1287615045931}{3504307716238}a^{7}+\frac{30018917441}{618407244042}a^{6}+\frac{18810943053991}{5256461574357}a^{5}+\frac{632687829338}{309203622021}a^{4}-\frac{2644119564465}{1752153858119}a^{3}+\frac{167093571927}{103067874007}a^{2}-\frac{35391052740779}{1752153858119}a-\frac{634271610083}{103067874007}$, $\frac{19398505999}{189232616676852}a^{15}+\frac{386508805}{5565665196378}a^{14}-\frac{96966863}{1590190056108}a^{13}+\frac{174884113}{1236814488084}a^{12}+\frac{5909617321}{399224929698}a^{11}-\frac{112622753}{11741909697}a^{10}+\frac{4947900166921}{31538769446142}a^{9}-\frac{73891340657}{618407244042}a^{8}-\frac{1287615045931}{3504307716238}a^{7}+\frac{30018917441}{618407244042}a^{6}-\frac{18810943053991}{5256461574357}a^{5}+\frac{632687829338}{309203622021}a^{4}+\frac{2644119564465}{1752153858119}a^{3}+\frac{167093571927}{103067874007}a^{2}+\frac{35391052740779}{1752153858119}a-\frac{634271610083}{103067874007}$, $\frac{99330041315}{189232616676852}a^{15}-\frac{8827785023}{11131330392756}a^{14}-\frac{6925318727}{5565665196378}a^{13}-\frac{21190547179}{11131330392756}a^{12}+\frac{14482705220}{199612464849}a^{11}+\frac{1283358509}{11741909697}a^{10}+\frac{2090759618569}{2252769246153}a^{9}+\frac{145182166750}{103067874007}a^{8}-\frac{427851446521}{5256461574357}a^{7}-\frac{3344674706}{44171946003}a^{6}-\frac{28991406029849}{1752153858119}a^{5}-\frac{2591567178554}{103067874007}a^{4}-\frac{34524010903646}{1752153858119}a^{3}-\frac{3211500018390}{103067874007}a^{2}+\frac{71727217254894}{1752153858119}a+\frac{888153176391}{14723982001}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 1376834470.0514348 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 1376834470.0514348 \cdot 1}{2\cdot\sqrt{321236373250909071617512439808}}\cr\approx \mathstrut & 0.484617191456913 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 144*x^12 - 1440*x^10 + 4392*x^8 + 31104*x^6 - 38880*x^4 - 171072*x^2 + 187272)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^16 - 144*x^12 - 1440*x^10 + 4392*x^8 + 31104*x^6 - 38880*x^4 - 171072*x^2 + 187272, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 144*x^12 - 1440*x^10 + 4392*x^8 + 31104*x^6 - 38880*x^4 - 171072*x^2 + 187272);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 144*x^12 - 1440*x^10 + 4392*x^8 + 31104*x^6 - 38880*x^4 - 171072*x^2 + 187272);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2^7.C_8$ (as 16T1155):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 1024 |
| The 40 conjugacy class representatives for $C_2^7.C_8$ |
| Character table for $C_2^7.C_8$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.173946175488.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | $16$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | $16$ | $16$ | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | $16$ | $16$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.79.2544 | $x^{16} + 32 x^{15} + 32 x^{14} + 40 x^{12} + 52 x^{8} + 48 x^{6} + 32 x^{5} + 32 x^{4} + 66$ | $16$ | $1$ | $79$ | 16T1155 | $[2, 3, 7/2, 4, 17/4, 19/4, 5, 41/8, 43/8, 6]$ |
|
\(3\)
| 3.16.12.3 | $x^{16} - 6 x^{12} + 162$ | $4$ | $4$ | $12$ | $C_{16} : C_2$ | $[\ ]_{4}^{8}$ |